Arbitrary Width Research Articles

(EXTRA)ORDINARY EQUIVALENCES WITH THE ASCENDING/DESCENDING SEQUENCE PRINCIPLE

AbstractWe analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C. Our main results are the following. The Rival–Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ over $\mathsf {RCA}_0$ . For each fixed $k \geq 3$ , the Rival–Sands theorem for infinite partial orders of width $\leq \!k$ is equivalent to $\mathsf {ADS}$ over $\mathsf {RCA}_0$ . The Rival–Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf {SADS}$ over $\mathsf {RCA}_0$ . Here $\mathsf {RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf {I}\Sigma ^0_{2}$ denotes the $\Sigma ^0_2$ induction scheme, $\mathsf {ADS}$ denotes the ascending/descending sequence principle, and $\mathsf {SADS}$ denotes the stable ascending/descending sequence principle. To the best of our knowledge, these versions of the Rival–Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ , by $\mathsf {ADS}$ , and by $\mathsf {SADS}$ . Furthermore, we give a new purely combinatorial result by extending the Rival–Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf {RCA}_0$ .

Rate dependent piezoresistive characterization of smart aerospace sandwich structures embedded with reduced graphene oxide (rGO) coated fabric sensors

In this study, the piezoresistive performance of smart aerospace sandwich structures was investigated by conducting three-point bending tests at variable loading rates (0.2, 2.0, and 20 mm/min). A reduced graphene oxide (rGO) coated glass fabric piezoresistive sensor was embedded in the facesheet of a sandwich structure. An integrated framework was designed to measure piezoresistive performance of the sandwich structure i.e. from fabrication of facesheets to electromechanical testing. The three-point bending tests performed on beams consisting of the coated fabric sensors with several different geometries (variable beam widths and core thicknesses) showed that the piezoresistivity of the fabric sensors was inversely proportional to the loading rate. Moreover, the study showed that the rate dependency was prominent in beams with larger widths having maximum percent decrease of 43% in the piezoresistivity (core thickness of 3 mm) with increasing the loading rate. However, the normalized piezoresistivity with width of the beam was found to be consistent for different widths, showing good application of the fabric sensor with any arbitrary width of interest. Furthermore, the piezoresistivity rate dependency was more prevalent in the non-linear region with a maximum of 68% decrease in piezoresistivity as compared to 42% found for the elastic region (width 75 mm, thickness 5 mm).

On occurrence disaggregation of probabilistic seismic hazard

AbstractDisaggregation of probabilistic seismic hazard allows to quantify how much one or more earthquake scenarios contribute to the occurrence [exceedance] of a ground motion intensity measure threshold of interest (x) at the construction site. The scenario is usually defined in terms of magnitude (M), source‐to‐site distance (R), and possibly includes the standardized residual (ε) of the ground motion model considered in the hazard analysis. Analytically, in case occurrence is of interest, disaggregation provides the joint probability density function of or conditional on the event, that is, or . Occurrence disaggregation is important for a number of earthquake engineering applications, and it is typically addressed in the literature in an approximated manner, considering as the conditioning event , with being an arbitrary finite width of the interval. This short communication undertakes a deeper examination of occurrence disaggregation clarifying that: (i) no approximation is needed in the case of disaggregation in terms of magnitude and distance (i.e., when is sought); (ii) is theoretically degenerate, and as such, its approximation via finite can lead to misleading results; (iii) if is chosen coherently with the discretization of the domain used in the hazard integral, it leads to approximated , enabling the conclusion that occurrence disaggregation does not add information with respect to disaggregation.

An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling

Map labeling is a classical problem in cartography and geographic information systems that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent (i.e., overlap-free) labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each timestep, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sublinear update time. Motivated by this, we study the maximal independent set ( MIS ) and maximum independent set ( Max-IS ) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types: (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining an MIS (and thus a 4-approximate Max-IS ) of a dynamic set of uniform rectangles with polylogarithmic update time. This breaks the natural barrier of \( \Omega (\Delta) \) update time (where \( \Delta \) is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes. For uniform rectangles, we first give an algorithm that maintains a 4-approximate Max-IS with \( O(1) \) update time. In a subsequent algorithm, we establish the trade-off between approximation quality \( 2(1+\frac{1}{k}) \) and update time \( O(k^2\log n) \) , for \( k\in \mathbb {N} \) . We conclude with an algorithm that maintains a 2-approximate Max-IS for dynamic sets of unit-height and arbitrary-width rectangles with \( O(\log ^2 n + \omega \log n) \) update time, where \( \omega \) is the maximum size of an independent set of rectangles stabbed by any horizontal line. We implement our algorithms and report the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances. We made several major observations in our empirical study. First, the original approximations are well above their respective worst-case ratios. Second, in comparison with the static approaches, the dynamic approaches show a significant speedup in practice. Third, the approximation algorithms show their predicted relative behavior. The better the solution quality, the worse the update times. Fourth, a simple greedy augmentation to the approximate solutions of the algorithms boost the solution sizes significantly in practice.

Semiclassical predictions of cosmological wave packets from ridgelines

We introduce a concept of ridgelines to investigate the semiclassical predictions from wave packets with arbitrary widths in conventional quantum mechanics and the Wheeler--DeWitt quantum cosmology. Two primary approaches are applied to the exact calculation of the ridgelines, namely the contour and the stream approaches. Moreover, aspects of these are discussed and compared to other scenarios and approaches, i.e., the narrow WKB wave packets and the first-derivative test. As the main result, we show that the semiclassical predictions in toy models have more abundant solutions than the ones in the classical theory, and most interestingly they may deviate from classical solutions due to the quantum corrections.

On the Optimization of Spread Spectrum Chirps Into Arbitrary Position and Width Pulse Signals. Application to Ultrasonic Sensors and Systems

This work presents in detail a new method for the optimization of rectangular spread spectrum excitation signals for its use in simple and low-cost ultrasonic pulsers, but that could be extended to other applications based on spread spectrum signals. Starting from rectangular linear frequency modulated (RLFM) chirps, it uses the transfer function of the transmitted signal and received echoes from reference specimens to iterate through a recursive algorithm to obtain arbitrary position and width pulse (APWP) signals with the desired bandwidth, maximizing the energy and the flatness of the spectrum, and enhancing the resolution and dynamic range of conventional chirp excitation signals. This optimization procedure can be repeated for any transducer and material, so that it achieves the best performance in each experimental environment. Such characteristics are ideal for Time-of-Flight estimation, imaging, and applications in which spectral regularity is needed, such as the Split Spectrum Processing (SSP) algorithm, which is used as example to test the performance of the proposed excitation signals. It also allows to specify and change the bandwidth reducing the need to change the transducers. The method is tested with different transducers (2 and 5 MHz focused transducers) and complex composite materials (aluminum-carbon aviation composite and high porosity GFRP composite) in immersion setup for imaging applications using SSP algorithm.

Design of the Ultra-Compact Silicon Multimode Waveguide Bends with Arbitrary Width and Radius Based on the Back Propagation Neural Network

Design of the Ultra-Compact Silicon Multimode Waveguide Bends with Arbitrary Width and Radius Based on the Back Propagation Neural Network

Artificial Intelligence-Assisted Design and Virtual Diagnostic for the Initial Condition of a Storage-Ring-Based Quantum Information System

Developments in Artificial Intelligence (AI) are helping to solve complex physical problems that otherwise may be too computationally demanding to solve using traditional approaches. Universal Approximation Theorems tell us that we can model any physical system if we can approximate the system with some continuous function (i.e., compact convergence topology and algorithmically generated sets of functions, such as the convolutional neural network), whether for an arbitrary depth or arbitrary width neural network. We consider the problem of solving a set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> coupled algebraic equations as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> becomes very large and apply machine learning (ML) to solve this problem for any value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> . The physical problem we are focusing on is to model the equilibrium positions of ions in an ion trap. A storage ring quantum computer could contain well over tens of thousands of ions. Quickly determining the equilibrium positions will be important to minimize the time to target and observe each ion. As each ion serves as a single qubit, this is important for setting and measuring the individual qubit states. The phonon modes from a collection of ions acts as another qubit, useful for gate operations. Measuring the phonon modes, where ions are oscillating around their respective equilibrium positions also means understanding the equilibrium positions very well. Turning all of this into a virtual diagnostic allows real time prediction and comparison to ensure unique definition of each ion.

Nonlinear Thomson scattering from a tightly focused circularly polarized laser with varied initial phases

Within the frame of classical electrodynamics, nonlinear Thomson scattering by an electron of a tightly focused circularly polarized laser has been investigated. The electron motion and spatial radiation characteristics are studied numerically when the electron is initially stationary. The numerical analysis shows that the direction of the maximum radiation power is in linear with the initial phase of the laser pulse. Furthermore, we generalize the rule to the case of arbitrary beam waist, peak amplitude and pulse width. Then the radiation distribution is studied when the electron propagates in the opposite sense with respect to the laser pulse and the linear relationship still holds true. Last we pointed out the limitation of the single electron model in this paper.

Noise and error analysis and optimization in particle-based kinetic plasma simulations

In this paper we analyze the noise in macro-particle methods used in plasma physics and fluid dynamics, leading to approaches for minimizing the total error, focusing on electrostatic models in one dimension. We begin by describing kernel density estimation for continuous values of the spatial variable x, expressing the kernel in a form in which its shape and width are represented separately. The covariance matrix C(x,y) of the noise in the density is computed, first for uniform true density. The bandwidth of the covariance matrix is related to the width of the kernel. A feature that stands out is the presence of constant negative terms in the elements of the covariance matrix both on and off-diagonal. These negative correlations are related to the fact that the total number of particles is fixed at each time step; they also lead to the property ∫C(x,y)dy=0. We investigate the effect of these negative correlations on the electric field computed by Gauss's law, finding that the noise in the electric field is related to a process called the Ornstein-Uhlenbeck bridge, leading to a covariance matrix of the electric field with variance significantly reduced relative to that of a Brownian process.For non-constant density, ρ(x), still with continuous x, we analyze the total error in the density estimation and discuss it in terms of bias-variance optimization (BVO). For some characteristic length l, determined by the density and its second derivative, and kernel width h, having too few particles within h leads to too much variance; for h that is large relative to l, there is too much smoothing of the density. The optimum between these two limits is found by BVO. For kernels of the same width, it is shown that this optimum (minimum) is weakly sensitive to the kernel shape.We repeat the analysis for x discretized on a grid. In this case the charge deposition rule is determined by a particle shape. An important property to be respected in the discrete system is the exact preservation of total charge on the grid; this property is necessary to ensure that the electric field is equal at both ends, consistent with periodic boundary conditions. We find that if the particle shapes satisfy a partition of unity property, the particle charge deposited on the grid is conserved exactly. Further, if the particle shape is expressed as the convolution of a kernel with another kernel that satisfies the partition of unity, then the particle shape obeys the partition of unity. This property holds for kernels of arbitrary width, including widths that are not integer multiples of the grid spacing.We show results relaxing the approximations used to do BVO optimization analytically, by doing numerical computations of the total error as a function of the kernel width, on a grid in x. The comparison between numerical and analytical results shows good agreement over a range of particle shapes.We discuss the practical implications of our results, including the criteria for design and implementation of computationally efficient particle shapes that take advantage of the developed theory.

Diffusion–reaction approach to electronic relaxation in solution: Exact solution of Smoluchowski equation for parabolic potential in presence of a rectangular sink

We give an analytical method for finding the exact solution for the problem of electronic relaxation in solution, modelled by a particle diffusing under the influence of parabolic potential in the presence of a sink of rectangular shape. The Smoluchowski equation is used to model the diffusive motion. We have derived an analytical formula for the average rate constant in closed form. Solving Smoluchowski equation analytically with a sink of arbitrary width has been an unsolved problem and is important as a realistic model for non-radiative electronic relaxation of a molecule in solution.

Analytical solution for nonlinear hydrologic routing with general power-law storage function

A new analytical solution for nonlinear storage routing with a general power-law storage function is presented. The resulting implicit solution, expressed in terms of the incomplete beta function, is valid for inflow hydrographs that may be represented by a series of pulses of arbitrary widths. The new solution thus extends the existing analytical solutions reported in the literature which are valid only for specific exponents in the power-law storage function. An explicit solution expressed in terms of the Lambert-W function is also presented for reservoirs for reservoirs with linear storage-elevation relationship and a single orifice or submerged sluice gate outlet. To support practical design applications and to demonstrate the utility of the new solution, we provide approximate explicit solutions and four simple applications. The results show that, being exact, the new solution is not subject to numerical errors or instabilities and may hence be used in a variety of modeling, design, optimization, forecasting, and control applications involving nonlinear routing with power-law storage functions.

ST-CAC: a low-cost crosstalk avoidance coding mechanism based on three-valued numerical system

Appearances of specific transition patterns during data transfer in bus lines of modern high-performance computing systems, such as communicating structures of accelerators for deep convolutional neural networks, commercial Network on Chips, and memories, can lead to crosstalk faults. With the shrinkage of technology size, crosstalk faults occurrence boosts and leads to degradation of reliability and performance, as well as the increasing power consumption of lines. One effective way to alleviate crosstalk faults is to avoid the appearance of these specific transition patterns by using numerical-based crosstalk avoidance codes (CACs). However, a serious problem with numerical-based CACs is their overheads in terms of required additional bus lines for representing code words. To solve this problem, in this paper we present a novel CAC that is based on the use of three symbols (three-value) to represent the code words in the bus lines, rather than classical binary CACs based on binary, i.e., 0 and 1 symbols. Our proposed CAC, named summation-based tri-value crosstalk avoidance code (ST-CAC), reduces the worst-case delay in bus lines with respect to binary CACs, and it can efficiently be applied to any arbitrary channel width of lines. The use of three symbols to represent code words in ST-CAC enables to increase the number of code words of a numerical system without increasing the number of required bus lines significantly. The experimental results show that CACs based on the use of three symbols can reduce the number of additional lines compared to binary CACs by 33%. Moreover, we show in the paper, that the delay of wires in the presence our ST-CAC can reduce by 33% with respect to state-of-the-art binary value CACs.

Compact Laser Driver for Ultrasonic Arbitrary Position and Width Pulse Sequences Generation

Laser ultrasound offers several benefits in comparison to immersion or air-coupled ultrasound: it directly generates stress on the sample surface; it has a wide bandwidth and a small acoustic source can be produced. Conventionally, high-power pulsed lasers are required therefore equipment is bulky and expensive. Here we propose to use mid-power laser diodes, which are small and low cost. To solve the low signal amplitude problem with lower power lasers, the use of spread spectrum signal, arbitrary position, and width binary pulse sets is proposed. To date, a laser driver for this task has not been available. This article describes the development of such a compact driver, where the essential electronics occupy an area of $10\,\,\times20$ mm. The whole system with a fixture for kinematic mount is comparable in size to a conventional piezoelectric ultrasonic transducer. The driver can supply pulse sets of up to 40 A. Individual pulse duration can vary between 20 and 1000 ns (0.5–25-MHz ultrasound range) and the total pulse train duration is limited by the laser type used. Experiments show that up to 10- $\mu \text{s}$ long pulse sets can be used at 10-A current, without laser degradation. Current waveforms, beam profile, and optical response signals were measured for three rectified topologies. It was concluded that the GaN-based constant current switch topology has the best performance, but a power MOSFET in a source current feedback topology can also be used to generate pulses down to 20 ns. Photoacoustic response signals from chirp and phase shift keying modulation have been demonstrated.

Deep Network Approximation for Smooth Functions

This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that multivariate polynomials can be approximated by deep ReLU networks of width $\mathcal{O}(N)$ and depth $\mathcal{O}(L)$ with an approximation error $\mathcal{O}(N^{-L})$. Through local Taylor expansions and their deep ReLU network approximations, we show that deep ReLU networks of width $\mathcal{O}(N\ln N)$ and depth $\mathcal{O}(L\ln L)$ can approximate $f\in C^s([0,1]^d)$ with a nearly optimal approximation error $\mathcal{O}(\|f\|_{C^s([0,1]^d)}N^{-2s/d}L^{-2s/d})$. Our estimate is non-asymptotic in the sense that it is valid for arbitrary width and depth specified by $N\in\mathbb{N}^+$ and $L\in\mathbb{N}^+$, respectively.

Boundary conformal field theory at the extraordinary transition: The layer susceptibility to O(\u03b5)

We present an analytic calculation of the layer (parallel) susceptibility at the extraordinary transition in a semi-infinite system with a flat boundary. Using the method of integral transforms put forward by McAvity and Osborn [Nucl. Phys. B455 (1995) 522] in the boundary CFT, we derive the coordinate-space representation of the mean-field propagator at the transition point. The simple algebraic structure of this function provides a practical possibility of higher-order calculations. Thus we calculate the explicit expression for the layer susceptibility at the extraordinary transition in the one-loop approximation. Our result is correct up to order O(ε) of the ε = 4 − d expansion and holds for arbitrary width of the layer and its position in the half-space. We discuss the general structure of our result and consider the limiting cases related to the boundary operator expansion and (bulk) operator product expansion. We compare our findings with previously known results and less complicated formulas in the case of the ordinary transition. We believe that analytic results for layer susceptibilities could be a good starting point for efficient calculations of two-point correlation functions. This possibility would be of great importance given the recent breakthrough in bulk and boundary conformal field theories in general dimensions.

Alternative quantisation condition for wavepacket dynamics in a hyperbolic double well

We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schrödinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system’s eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication Chomet et al (2019 New J. Phys. 21 123004). Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.

Analytical solution in Laplace domain of Smoluchowski equation for a flat potential with a rectangular sink

We propose a new method for finding the exact analytical solution in Laplace domain for the problem where a particle is diffusing on a flat potential in the presence of a rectangular sink of arbitrary width and height. In our model, diffusive motion is described by the Smoluchowski equation. Our method with this sink of rectangular shape is very general and can be used to deal with other potentials. We have derived exact analytical expression for rate constants using our model. This is the first model where the exact analytical solution in closed form is found in the case of a sink of arbitrary width. This model is more realistic for understanding reaction–diffusion systems that all other existing models available in literature.

Interaction of flexural gravity wave in ice cover with a pair of bottom-mounted rectangular barriers

This paper considers the reflection and transmission of flexural gravity waves by a pair of bottom-mounted rectangular barriers of arbitrary height and width. Depth of the fluid considered here is finite but not necessarily uniform. The fluid region is bounded by an infinite long thin ice sheet of homogeneous properties. Using linearized water wave theory the velocity potential associated with the motion is formulated. Eigenfunction matching technique is used to convert the boundary value problem to a set of integral equations which are solved by using multi-term Galerkin method. The reflection and transmission coefficients are calculated and represented graphically. An expression for the energy balance relation is given and checked carefully. A model of the problem in which the fluid is bounded by a free surface is recovered. For a set of physical parameters the results are analyzed through a number of figures. The convergence and correctness of the method is confirmed by comparing the results with the one available in the literature. It is shown that a pair of thick bottom-mounted barriers have significant impact on reflected and transmitted flexural gravity waves. Resonance effects are observed and verified for different parameters of the current problem.

Quantification of Joint Space Width Difference on Radiography Via Phase-Only Correlation (POC) Analysis: a Phantom Study Comparing with Various Tomographical Modalities Using Conventional Margin-Contouring.

Several visual scoring methods are currently used to assess progression of rheumatoid arthritis (RA) on radiography. However, they are limited by its subjectivity and insufficient sensitivity. We have developed an original measurement system which uses a technique called phase-only correlation (POC). The purpose of this study is to validate the system by using a phantom simulating the joint of RA patients.A micrometer measurement apparatus that can adjust arbitrary joint space width (JSW) in a phantom joint was developed to define true JSW. The phantom was scanned with radiography, 320 multi detector CT (MDCT), high-resolution peripheral quantitative CT (HR-pQCT), cone beam CT (CBCT), and tomosynthesis. The width was adjusted to the average size of a women's metacarpophalangeal joint, from 1.2 to 2.2mm with increments of 0.1mm and 0.01mm. Radiographical images were analyzed by the POC-based system and manual method, and images from various tomographical modalities were measured via the automatic margin detection method. Correlation coefficients between true JSW difference and measured JSW difference were all strong at 0.1mm intervals with radiography (POC-based system and manual method), CBCT, 320MDCT, HR-pQCT, and tomosynthesis. At 0.01mm intervals, radiography (POC-based system), 320MDCT, and HR-pQCT had strong correlations, while radiography (manual method) and CBCT had low correlations, and tomosynthesis had no statistically significant correlation. The smallest detectable changes for radiography (POC-based system), radiography (manual method), 320MDCT, HR-pQCT, CBCT, and tomosynthesis were 0.020mm, 0.041mm, 0.076mm, 0.077mm, 0.057mm, and 0.087mm, respectively. We conclude that radiography analyzed with the POC-based system might sensitively detect minute joint space changes of the finger joint.